Let vj (vj1, ,Vjn) Rn, j 1, , n, be fixed The parallelepiped determined by the vectors vj is the set and the determinant of the vj 's is the number det(v1 ,vn) del vjk n n Prove that Vol(P(v1 ,vn)) det(v1, ,vn) Check this formula for n 2 and n 3 to see that it agrees with the classical formulas for the area of a parallelogram and the volume of a parallelepiped P(v1 Vn) (Vi I, Vn tj 0, 1 ),

Question: Let vj = (vj1,. . . . . . . . .,Vjn) Rn, j = 1,. . . . . . ., n, be

Let vj = (vj1,. . . . . . . . .,Vjn) ˆˆ Rn, j = 1,. . . . . . ., n, be fixed. The parallelepiped determined by the vectors vj is the set

and the determinant of the vj's is the number
det(v1...,vn) := del[vjk]n Ã— n.
Prove that
Vol(P(v1...,vn)) = |det(v1,...,vn)|.
Check this formula for n = 2 and n = 3 to see that it agrees with the classical formulas for the area of a parallelogram and the volume of a parallelepiped.

P(v1..... Vn) = (Vi ++ I, Vn : tj [0, 1]),

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